Concerning the generation of geomagnetic giant pulsations
by drift-bounce resonance ring current instabilities
K.-H. Glassmeier1, S. Buchert1, U. Motschmann1, A. Korth2, A. Pedersen3
1Institute for Geophysics and Meteorology, Technical University of Braunschweig, Germany 2Max-Planck-Institute for Aeronomy, Katlenburg-Lindau, Germany
3European Space Research and Technology Centre, Noordwijk, The Netherlands Received: 7 July 1997 / Revised: 19 August 1998 / Accepted: 21 August 1998
Abstract. Giant pulsations are nearly monochromatic ULF-pulsations of the Earth's magnetic ®eld with periods of about 100 s and amplitudes of up to 40 nT. For one such event ground-magnetic observations as well as simultaneous GEOS-2 magnetic and electric ®eld data and proton ¯ux measurements made in the geostationary orbit have been analysed. The observa-tions of the electromagnetic ®eld indicate the excitation of an odd-mode type fundamental ®eld line oscillation. A clear correlation between variations of the proton ¯ux in the energy range 30±90 keV with the giant pulsation event observed at the ground is found. Furthermore, the proton phase space density exhibits a bump-on-the-tail signature at about 60 keV. Assuming a drift-bounce resonance instability as a possible generation mecha-nism, the azimuthal wave number of the pulsation wave ®eld may be determined using a generalized resonance condition. The value determined in this way, m ÿ214, is in accord with the value m ÿ276 determined from ground-magnetic measurements. A more detailed examination of the observed ring current plasma distribution function f shows that odd-mode type eigenoscillations are expected for the case @f=@W >0, much as observed. This result is dierent from previous theoretical studies as we not only consider local gradients of the distribution function in real space, but also in velocity space. It is therefore concluded that the observed giant pulsation is the result of a drift-bounce resonance instability of the ring current plasma coupling to an odd-mode fundamental standing wave. The generation of the bump-on-the-tail distribution causing @f=@W >0 can be explained due to velocity dispersion of protons injected into the ring current. Both this velocity dispersion and the necessary substorm activity causing the injection of protons into the nightside magnetosphere are observed.
Key words. Magnetospheric physics (energetic particles, trapped; MHD waves and instabilities)áSpace plasma physics (wave-particle interactions).
1 Introduction
Giant pulsations (Pg) are ULF-pulsations predominant-ly occurring in the morningside magnetosphere. They are characterised by their large amplitudes, and their striking quasi-monochromatic wave form with periods of typically 100 s. Giant pulsations have been studied intensively in the past by e.g. Annexstad and Wilson (1968), Green (1979), Rostokeret al. (1979), Glassmeier (1980), and more recently by Chisham and Orr (1991), Chishamet al. (1992), Takahashiet al. (1992), Chisham and Orr (1994), Chisham (1996), and Chisham et al. (1997).
These studies show that giant pulsations exhibit a strongly localised pulsation wave ®eld extending only 200±300 km in the north-south direction, and 600±700 km in the east-west direction at the surface of the Earth. Furthermore, Pgs are found to be characterised by rapid east-west phase variations with the azimuthal wave number, m, reaching values up to m ÿ40, where the negative sign indicates westward phase propagation. These characteristics led several authors to conclude that giant pulsation wave ®elds are propagated in the guided poloidal mode, ®rst described in Dungey's (1954) seminal work on ULF-pulsation theory. Although Pg ground amplitudes are often smaller than those of Pc4-5 pulsations the ionospheric magnetic ®eld of Pgs is much larger due to the screening eect of the atmosphere which scales exponentially with the azimuthal wave number (e.g. Hughes and Southwood, 1976; Glassmeier, 1984). This justi®es calling this type of pulsation giant pulsations.
However, there is still considerable debate about the possible generation mechanism of Pgs. The fast Correspondence to:K.-H. Glassmeier
azimuthal phase variation and indications toward an odd-mode wave structure led Green (1979), Takahashi
et al. (1992), and others to suggest that a drift-wave instability of the magnetospheric ring current (e.g. Hasegawa, 1971) is the dominant plasma instability generating giant pulsations. Other authors such as Glassmeier (1980), Poulter et al. (1983), or Chisham and Orr (1991) favour the proton drift-bounce reso-nance instability (e.g. Southwood et al., 1969; South-wood, 1976) as a generation mechanism, and give arguments for an even-mode wave structure along geomagnetic ®eld lines. However, no ®rm conclusion on the generation mechanism has been reached as yet.
Any future theory on the generation mechanism of giant pulsations requires solution of the following three problems: ®rst, identi®cation of the instability mecha-nism; second, identi®cation of the mechanism generat-ing the unstable plasma situation, and providgenerat-ing the plasma free energy to feed the instability; third, identi-®cation of the wave mode the unstable plasma couples to, and by which the disturbance is propagated within the magnetosphere, especially to the ground. Further-more, any theory should be able to predict and explain the striking quasi-monochromatic wave form, the fast azimuthal phase variation, the strong spatial localisa-tion, and the predominantly morningside occurrence.
Using correlated ground-magnetic and geostationary satellite observations of a giant pulsation event which occurred on October 4, 1978, 0645-0830 UT (0915-1100
MLT), these mentioned problems will be tackled and a conjecture is formulated which explains many typical characteristics of giant pulsations.
2 Ground-magnetic observations
During the period of the International Magnetospheric Study, 1976±1979, the University of MuÈnster operated the Scandinavian Magnetometer Array (Fig. 1; for further details see KuÈppers et al., 1979), whose Earth magnetic ®eld observations will be used in the present study. Figure 2 displays bandpass ®ltered (60 s <T <300 s) records of the D-component of the Earth magnetic ®eld variations in the time interval October 4, 1978, 0645-0830 UT along a north-south pro®le, pro®le 4 (see Fig. 1) of the Scandinavian magnetometer array (SMA). The station SOY at about 71 geomagnetic latitude is the northernmost, and OUL
at about 63latitude the southernmost station along the
pro®le; station separation is about 120 km.
From the data in Fig. 2 it can be seen that clear Pg-activity with a period of about 100 s was recorded in Northern Scandinavia in the time intervals 0650-0718 UT ( MLT UT 2:5 h), and 0735-0818 UT. The Pg is most prominent at the stations MUO and PEL, where peak-to-peak amplitudes reach values of up to 20 nT. To the north and south of these two stations the Pg becomes less visible and more irregular, especially at the
northern stations. This hints toward the usually ob-served strong NS-localisation of the Pg-wave ®eld (e.g. Glassmeier, 1980).
The observations indicate a clear amplitude modula-tion, or better, a double-event characteristic of the giant pulsation with amplitude maxima at 0658 UT and around 0755 UT. Between the two wave groups a short-duration Pc5 event may be identi®ed at about 0720 UT. We do not regard this Pc5 event as associated with the Pg-phenom-ena and do not discuss it any further. The clear amplitude modulation allows us to determine the horizontal group velocity. However, in contrast to earlier work by Glass-meier (1980) or Poulter et al. (1983), where westward velocities in the Earth's frame of reference of the order of 200 m/s were found, inspection of the Pg signal along east-west pro®les AND-MAT-VAD, FRE-EVE-ROS-MIE-SKO, and GLO-RIJ-KIR-MUO (see Fig. 1) indi-cates no appreciable east-west group motion in the Earth's frame of reference for the Pg event studied.
The azimuthal phase velocity can be determined as displayed in Fig. 3, where records of the D-component are shown along an east-west station pro®le RIJ-KIR-MUO (see Fig. 1) for the time interval 0652-0658 UT. The pro®le is located at about 65geomagnetic latitude.
The oscillation maximum is ®rst seen at the easternmost station MUO and about 30 s later at the westernmost station RIJ. Thus, phase propagation is from east to west in the Earth's frame of reference. The apparent horizontal phase speed is vph ÿ0:135 deg/s, where the
negative sign denotes westward propagation, that is variations in azimuth are counted positive eastward. This corresponds to an apparent angular phase velocity
around the Earth ofXA ÿ2:410ÿ3 rad/s. With the
Pg-frequencyx2p=100sthis gives an azimuthal wave number m ÿ267, with the error estimate resulting from a sampling period of 10 s for the magnetic ®eld observations used.
3 Satellite observations
For the giant pulsation event analysed satellite obser-vations of the magnetospheric electromagnetic ®eld as well as electron and ion ¯ux measurements made onboard the European geostationary satellite GEOS-2 are available. GEOS-2 was launched into a geostation-ary orbit on July 14, 1978. During the time interval discussed the satellite was located at 37 eastern
geographic longitude; local time onboard GEOS-2 corresponds to UT+150 min. Field line tracing under moderate Kp conditions with a Tsyganenko-magneto-spheric magnetic ®eld model gives a nominal footpoint of the GEOS-2 ®eld line close to the SMA station ROS (see Fig. 1). Thus the SMA and GEOS-2 are under nominal conjugate conditions. The geomagnetic longi-tude of GEOS-2 in the equatorial plane is determined as 104.
Having available satellite electric and magnetic observations is a signi®cant achievement in trying to clarify the Pg generation mechanism. For example, inspection of the electromagnetic ®eld observations allows us to determine the mode of oscillation of the observed event. Previous studies reached con¯icting results with respect to this question. Green (1979), and more recently Takahashi et al. (1992) favour an odd-mode, fundamental oscillation with anti-symmetric (symmetric) transverse magnetic ®eld (electric ®eld and ®eld line displacement) variations. On the other hand, Glassmeier (1980), and Chisham and Orr (1991) give arguments for and strongly favour an even-mode, second harmonic standing ®eld line oscillation with symmetric (anti-symmetric) transverse magnetic ®eld (electric ®eld and ®eld line displacement) variations. An
Fig. 2. Magnetic ®eld variations recorded along pro®le 4 of the Scandinavian magnetometer array for the time interval October 4, 1978, 0645-0830 UT (0845-1030 MLT). The data have been band-pass ®ltered in the period range 60±300 s
even mode (odd mode) can thus be identi®ed in the magnetospheric equatorial plane as an oscillation ex-hibiting a node (an anti-node) of the electric ®eld oscillation and an anti-node (a node) of the transverse magnetic ®eld variation.
Figure 4 displays magnetic and electric ®eld obser-vations made onboard GEOS-2 in the time interval October 4, 1978, 0600-0800 UT. Magnetic ®eld data are taken from the GEOS-2 ¯uxgate magnetometer (Can-didi et al., 1974) and are represented in a dipole coordinate system. Thus, Br denotes the radial
compo-nent, counted positive outward, B/ the azimuthal
component, counted positive to the east, and Bh
completes the right-hand system. In all three magnetic ®eld components no clear signal resembling, in any way, the characteristics of the giant pulsation event recorded at the ground can be identi®ed. The regular, about 1 min oscillations in the Br and B/ components are
instru-mental and due to a beat between the spin rate of 6 s and the data readout time of 5.5 s. These oscillations do not indicate any natural variation. It is worthwhile noting that only the magnetic ®eld magnitude increases (seen in the Bh component) at about 0615 and 0630 UT.
The electric ®eld observations, however, display clearly the giant pulsation seen at the ground. The coordinate system used is again a dipole coordinate
system with Ehapproximating the magnetic ®eld-aligned
component. It does not completely vanish due to deviations from theh-direction from the actual magnetic ®eld direction. Spiky perturbations at about 4 min past each hour are due to calibration sequences. The main Pg signal is recorded in the azimuthal component. At about 0630 UT a wave packet is seen lasting until about 0700 UT, when activity ceases for about 10 min. From 0707 UT until 0737 UT the event is seen again. Some activity is also seen in the radial component. Spectral analysis reveals a pulsation period ofT 1105s.
The correspondence between the period of the pulsation seen at the ground and in space as well as the clear packet structure of the event with almost the same packet length (see Figs. 2 and 4) strongly suggests that the same giant pulsation event has been observed. Moreover, it should be noted that the event at GEOS-2 already starts at 0630 UT, that is about 15 min earlier than at the ground. This is somewhat surprising as the nominal footpoint of the GEOS-2 ®eld line is covered by the SMA viewing area. Thus, simultaneous onset of the perturbation was expected. We have, at present, no other explanation for this dierence in onset time than questioning the validity of the ®eld line tracing method used. As can be seen from Fig. 4 the magnetospheric magnetic ®eld signi®cantly changed at about 0630 UT,
a temporal variation which cannot be covered by any means with a Tsyganenko model. As the geographic position of GEOS-2 is about 20 east of Scandinavia it
is justi®ed assuming that the true footpoint of the GEOS-2 ®eld line is also located east of Scandinavia. Glassmeier (1980) and Poulter et al. (1983) reported westward group velocities of 200±300 m/s. Thus a delay time of 15 min between ground and satellite can easily be explained by an eastward shift of the footpoint of a few hundred kilometres. If the true footpoint is located east and if an azimuthal westward group velocity of a few hundred metres is assumed a delay of 15 min is easily explained.
The observation of transverse electric ®eld variations in the magnetospheric equatorial plane and the absence of any correlated magnetic ®eld variations indicates the presence of an odd mode standing ®eld line oscillation. Whether the fundamental or any odd harmonic oscilla-tion has been excited can be decided on grounds of the wave period. The electron density was measured at about 3 cmÿ3; assuming a plasma predominantly
consisting of protons gives an equatorial mass density of 510ÿ21 kg/m3. The observed large azimuthal wave
number m suggests that a guided poloidal mode has been excited. For this type of mode Cummings et al. (1969) have calculated the eigenperiods for dierent harmonics and density indices n for density pro®les
q r q0 r0=rnalong ®eld lines passing the
geostation-ary orbit, r06.6 RE. With this value forq0510ÿ21
kg/m3and n = 1 eigenperiods of T = 127 s and T = 23
s for the fundamental and the third harmonic guided poloidal mode, respectively, result. We may conclude that the observed electric ®eld observations are due to a fundamental mode poloidal eigenoscillation. The ob-served predominantly azimuthal electric ®eld variation corresponds to a predominantly radial magnetic ®eld perturbation. Taking into account an ionospheric rota-tion by 90 (e.g. Hughes, 1974; Glassmeier, 1984) a
ground-magnetic perturbation dominating the D-varia-tion is expected. This is indeed observed, which supports our conjecture of a guided poloidal mode having been excited.
However, the observed electron density may be too low because only part of the distribution function has been measured. Furthermore, the magnetospheric plas-ma is not a pure electron-proton plasplas-ma, but can also exhibit a signi®cant oxygen component. Wedekenet al. (1984), for example, found a ratio of oxygen to proton ions of 1/15 at the geostationary orbit. Singer et al. (1979) report even higher ratios. Using the Wedeken
et al. (1984) value a mass density of q06:4
10ÿ21kg/m3 results. This corresponds to poloidal mode
eigenperiods of 145 s, 41 s, and 26 s for the fundamental, second harmonic, and third harmonic mode, respec-tively. A clear identi®cation of the mode structure is thus not possible due to a lack of suitable mass density measurements.
Electron and proton observations made onboard the GEOS-2 satellite have been studied to ®nd correlative signatures with the ground magnetic and the satellite electric ®eld oscillations, identi®ed as giant pulsations.
The particle measurements were performed with a magnetic spectrometer to separate electrons from ions (Korth and Wilken, 1978; Korthet al., 1978). Electron energy spectra are determined in 16 channels between 16 keV and 213 keV. Ions are analysed in 10 energy channels between 28 and 402 keV. Full energy spectra are obtained every 5.5 s in various pitch angle ranges. No identi®cation of dierent ion species is possible. For the data interpretation we assume that most of the detected ions are protons.
From the data analysis of the particle instrument follows that only the ion observations signi®cantly correlate with the giant pulsation event observed. Figure 5 displays proton ¯ux measurements in six dierent energy channels from 28 keV up to 114 keV in the pitch angle range 25ÿ35. The ¯ux shows appreciable
variations in the dierent energy ranges. The most pronounced variations are increases of the ¯ux at around 0700 UT and 0800 UT (see, for example, channel 5, 59±75 keV, in Fig. 5). These ¯ux intensi®ca-tions parallel the amplitude modulation of the Pg as observed in the ground-magnetic ®eld and the GEOS-2 electric ®eld observations. This indicates a possible relationship bewteen the modulation and ¯ux increases. A more detailed interpretation, however, is not yet possible as the ¯ux intensi®cations are local eects while the wave activity is caused by local plasma instabilities as well as waves which propagate in from other regions of the magnetosphere to the points of observations. Azimuthal Poynting ¯ux observations are required to elucidate this point further. They are not available for the present study.
The proton ¯uxes also show velocity dispersion, that is the higher-energy protons arrive earlier than the lower-energy ones. This points toward the proton injection not occurring locally, but at some distance from the satellite. It should be noted that this modula-tion feature of the proton ¯uxes as well as the velocity dispersion can be observed for protons with pitch-angles up to 40, but not at higher pitch-angles, for which no
correlation between proton ¯ux and ground-magnetic ®eld variations has been found. Also, the proton ¯ux observations do not indicate the presence of a signi®cant proton temperature anisotropy which could give rise to a drift-mirror instability.
A possible injection mechanism for the observed proton clouds is substorm activity in the nightside magnetosphere. Figure 6 displays ground-magnetic ®eld variations observed at the Canadian station Fort Prov-idence (located at 292geomagnetic longitude) during the
interval October 4, 1978, 0400±0900 UT (2100±0200 MLT), and indicates several substorms or substorm intensi®cations in the nightside magnetosphere at the same time the giant pulsation event is observed in the morningside (see Figs. 2 and 6). Substorm activity is therefore a candidate for the observed proton injections. Therefore, a relation between substorm activity and Pgs is suggestive, and such a connection has also been discussed by Glassmeieret al. (1992) and Chishamet al. (1992).
and also the drift periods of the protons. If / is the actual longitude a proton has reached after the timetat the injection point/0, the relation
/ t /0 v/t 1
holds where v/ is the azimuthal drift velocity. This
velocity is negative for protons due to our de®nition of azimuthal variations counted positive eastward. Ac-cording to Schulz and Lanzerotti (1974) v//W, the
particle energy. With the dierence in arrival time,
dttiÿtj, observed at the same place between the
particles in energy channels i and j, from the earlier equation
tj WWidt
jÿWi 2
where for the particle energies Wj > Wi. The velocity
dispersion as seen in Fig. 5 is emphasized most in the energy channels 3±6. With Eq. (2) a drift time of e.g. t52935 s for the 59±75 keV protons results.
Assuming an injection point at around midnight (that is when Fort Providence exhibits substorm activity) the protons observed at GEOS-2 have thus drifted a distance of about ÿ188 (the distance between Fort
Providence and GEOS-2 in geomagnetic longitude) within this drift time. Accordingly a drift frequency
XD;5 ÿ 1:10:2 10ÿ3radsÿ1 may be determined
from the particle observations. Here we allowed for an error of about 20% in the determination of XD to account for the uncertainty of the determination of the injection point.
In a dipole ®eld geometry gradient and curvature drift for 67 keV protons would result in a drift frequency
XD ÿ0:810ÿ3radsÿ1. This value corresponds to the
measured one. The proton ¯ux measurements can be used to derive phase space densities as well. Figure 7 exhibits phase space densities at times 0655:18 and 0700:21 UT for the pitch angle range 5ÿ15. The most
remarkable feature is a shoulder at an energy of about 60 keV, i.e. the energy range of channel 5. A similar shoulder is also seen at the same energy in other pitch angle ranges below 40. We interpret this shoulder as
indicating an unstable, bump-on-the-tail distribution. Thus, not only the correlation between Pg amplitude modulation and proton injection, but also the phase space densities point toward a causal relationship between the observed giant pulsations and proton injection.
4 Drift-bounce resonance instability of the ring current plasma
Hughes et al. (1978) reported a quasi-monochromatic wave event observed onboard the satellite ATS 6 in the afternoon sector of the magnetosphere and concurrent measurements of bump-on-the-tail protons phase space distributions. They convincingly argue that this unstable distribution via the bounce-resonance instability gives rise to the observed quasi-monochromatic wave. How-ever, their bump-on-the-tail was observed at an energy of about 10 keV, and the wave was not identi®ed as a giant pulsation.
Nevertheless, it is tempting to discuss the described GEOS-2 proton observations in terms of a drift-bounce resonance instability of the ring current plasma (South-woodet al., 1969). This instability might be discussed as an analogy of the cyclotron instability. In a mirror ®eld
geometry a charged particle performs a bounce motion between the mirror points and an azimuthal drift motion in addition to the gyromotion around the local magnetic ®eld direction. While the cyclotron instability is a resonant interaction between a plasma wave propagat-ing along the ambient magnetic ®eld and the gyratpropagat-ing particles, the drift-bounce resonance describes a reso-nant interaction between the drift-bounce motion of the particles and an azimuthally propagating wave. The change of the energy, W, of the particle with time is given by
dW
dt q vkEk l @bk
@t q~E~vD 3 where qand l are the electric charge and the magnetic moment of the particles, respectively, and where vk; bk,
and Ek denote the particle velocity along the ambient
magnetic ®eld as well as the wave magnetic and electric ®eld perturbations along the background ®eld, and~vDis
the unperturbed particle drift velocity. If the particle interacts with a purely transverse magnetohydrodynam-ic wave the ®rst two terms in Eq. (3) vanish, and the interaction is via the third term only, i.e. in an inhomogeneous background ®eld the azimuthal gradient and curvature drifts cause an interaction with the azimuthal wave electric ®eld.
Fig. 6. Magnetic ®eld observations, recorded on October 4, 1978, 04± 09 UT, at Fort Providence, Canada (data courtesy G. Rostoker)
The resonance condition is given by (e.g. Southwood, 1976)
x ÿ mXDÿNXB 0 4
where x is the wave frequency, m an azimuthal wave number, and XD and XB denote the particle drift and bounce frequencies, respectively. According to previous studies on the bounce resonance instability by e.g. Southwoodet al. (1969), Southwood (1976), Glassmeier (1980), Southwood and Kivelson (1982), or Takahashi
et al. (1992)N is an integer. However, this is only true if the transverse component of the electric ®eld perturba-tion is symmetric or antisymmetric about the magnetic equator. Any asymmetry allows N to take any real value. Such an asymmetry can be due to asymmetry of the ionospheric Pedersen conductivity between the Northern and Southern Hemisphere. This occurs, for example, at the solstices, when one polar ionosphere is illuminated for a long time and the other is correspond-ingly in darkness. In this case the resulting standing wave has a quarter-wave form along the ®eld line with clear asymmetry (Allan and Knox, 1978).
In more general terms the wave ®eld along the ambient magnetic ®eld must be regarded as a superpo-sition of the ®elds of a standing wave and a travelling wave. Finite ionospheric conductivity causes a ®nite Joule heating in and thus a Poynting ¯ux into the ionosphere (e.g. Allan and Knox, 1979; Glassmeier
et al., 1984). If the Pedersen conductivities are equal in both hemispheres the Poynting ¯ux into the iono-spheres, integrated along the ®eld line, must also be equal. This requires the existence of a null-point, where the local Poynting ¯ux vanishes (Allan, 1982). For symmetric or antisymmetric electric and magnetic ®eld perturbations this null-point is located right at the magnetic equator. However, if the ionospheric conduc-tivities dier from each other this null-point is signi®-cantly shifted towards that ionosphere where the conductivity is larger. This implies a break-down of any symmetric or antisymmetric behaviour of the wave ®elds along the ®eld line with respect to the magnetic equator. The null-point may be regarded as the new point of symmetry.
The eect of asymmetries on the resonance condition (4) can be demonstrated by the following simpli®ed computation of the energy exchange per time between the wave ®eld and a bouncing particle. The energy change of a particle due to its interaction with the electromagnetic ®eld of the unstable wave is given by
dW_B s qE/ svD s exp i m/ÿxt 5
where qis the electric charge of the particle, E/ s the
arc length s dependant azimuthal electric ®eld compo-nent of the wave, vD s the arc length dependant azimuthal drift velocity, and / the azimuth angle or azimuthal drift phase. The electric ®eld may be written as
E/ s ÿiEo exp
iap L s
6
This expression allows for asymmetric electric ®eld variations with respect to the equatorial plane. Param-etersaandLdescribe this asymmetry and the arc length between the northern and southern footpoint of the oscillating ®eld line, respectively. The arc length is counted positive from the northern mirror point. For
a1 this functional dependance of E/ allows to
describe the odd-mode fundamental eigenoscillation. The quarter-wave is described by a1=2, while e.g.
a2=3 describes an electric ®eld variation which is asymmetric with respect tosL=2 and exhibits dier-ent values of the electric ®eld at conjugate ionospheres. Such a mode would allow us to describe a situation with the null-point shifted towards one hemisphere.
For a particle bouncing in a dipole ®eld one has
sl cosXBt 7
with lthe half-bounce path length, that is the recti®ed path distance between the northern and southern mirror points of the bouncing particle (e.g. Roederer, 1970, p.35);s0 for the northern mirror point. This relation between arc length of the bouncing particle and time re¯ects the harmonic motion of the particle in a dipole ®eld, provided cos2h1, where the angle hXBt
denotes the particle bounce phase (e.g. Schulz, 1991, p. 203). The general expression relating arc length and bounce phase is a more complicated nonlinear relation given by Eq. (150) of Schulz (1991). With this expression for the arc lengths the variation of the electric ®eld as seen by a bouncing particle can be approximated by
E/ s ÿiEo exp iapLl cosXBt
8
The arc length dependant drift velocity may be approx-imated by its bounce averaged value via
vD s hvD si vD 9
and the drift phase is given by/ t XDt. With these de®nitions one has
dW_B ÿiEovDexp
i mXDÿxtiapLlcosXBt
10
In this expression wave frequencyxand drift frequency
XDappear in a linear relationship with timet, while the bounce frequency XB is related to time in a nonlinear cosine function term. To obtain a similar linear relation between bounce frequency and time as one has between drift frequency and time the expression exp iapl
L cosXBt
may be expanded into a Fourier series as done by e.g. Southwood (1976):
expiapl L cosXBt
NX1 Nÿ1
cN exp iNXBt 11
It should be noted that Southwood (1976) actually expands the bounce phaseh. The expansion coecients
cN denote the spectral weight of each bounce phase
Of particular interest is the bounce integrated energy transfer, that is;
dWB Z TB
0 d
_
WBdt 12
with the bounce period TB2p=XB. Southwood's
(1976) expansion results in:
dWB qEovD X N1
Nÿ1
cNexp i xmÿXDmÿXDxÿNNXBXB t TB
0 ;
13
an expression approximated by Southwood [1976] as:
dWB qEovDd xÿmXDÿNXB exp i mXDÿxNXBtTB
0 : 14
It is at this point where the condition N 2N has been introduced by truncating the series over N. This truncation implies the following approximation: exp iapLlcosXBt
exp iNXBt: 15
A dierent approximation allows relaxation of the condition N 2N. The cosine dependence s(t) may be approximated by a triangle function via:
s t plXBt if 0tTB=2 l
p 1ÿXBt if TB=2tTB: (
16
For tTB=2 this gives, for example, s tTB=2 l:
This triangular approximation provides one with a linear relationship between s and t. It thus equals the already used relations between frequency and time as well as drift frequency and time. No expansion as used by Southwood (1976) is required. With this approxima-tion the bounce integrated energy change reads:
dWB ÿiqEovD
Z TB=2
0 exp i mXDnXBÿx tdt
Z TB
TB=2
exp i mXDÿnXBÿx t
exp indt
17
where nal=Lhas been introduced. Integration of the ®rst term on the right hand side of Eq. (17) gives
Z TB=2
0 exp i mXDnXBÿx tdt
xÿmXDi ÿnXBexp i mXDnXBÿx t TB=2
0 : 18
This expression maximizes, if xÿmXDÿnXB0. Integration of the second term on the right hand side of Eq. (17) gives
Z TB
TB=2
exp in exp i mXDÿnXBÿx tdt
xÿi expmXD innXBexp i mXDÿnXBÿx t TB
TB=2
:
19
This expression maximizes, if xÿmXDnXB0. Thus, the resonance condition reads
xÿmXDÿnXB 0 20
with n2R. This is a proper generalization of the Southwood (1976) condition. Eqs. (4) and (20) agree, if Lland ifa 1;2. . .
This relation can be used to determine the azimuthal wave number
m xÿnXB
XD 21
wherem> <0 corresponds to an eastward (westward) propagating wave and a suitable value for nhas to be assumed. The protons under consideration have a pitch angle of about 30, that is they mirror at an equatorial
distance of about 35 or l/2 = 0.25 L. Thus, we have
l=L0:5.
To estimate the value ais more dicult. It requires knowledge of the ionospheric conductivity at the foot-points of those ®eld lines perturbed by the Pg pulsation considered. Measurements of the conductivities are not available. However, using a Tsyganenko 89 magnetic ®eld model to determine the ionospheric footpoints and the IRI 90 ionosphere model values for the Pedersen conductance of 3.3 S and 8.4 S in the Northern and Southern Hemisphere, respectively, are found. The clear dierence in hemispheric conductances indicates a clear shift of the null-point toward the Southern Hemisphere. This asymmetry of the Pg electric ®eld variation may be described by a value a0:8. Note, that a1 would correspond to a fundamental mode oscillation. With these estimates forl=Land awe haven0:4.
The bounce frequency may be computed using the relation (Schulz and Lanzerotti, 1974)
XB p
2mp p
L RE
W
p
1
T y 22
wheremp is the particle mass, RE= 6371 km the Earth
radius,Lthe McIlwain parameter,W the particle energy, andT ya factor taking into account the pitch angle a
with ycosa. For the pitch angles discussed here T y 1:3. Thus with W 67keV, a value
XB10ÿ1radsÿ1 results. With x2p=100 s, where
T 100s is the observed giant pulsation period, and
XD ÿ 1:10:2 10ÿ3radsÿ1, the empirically
deter-mined channel 5 proton drift frequency, we receive azimuthal wave numbers
m ÿ21 4
for n 0:4, respectively. The error estimate is based on the error in the determination ofXD. Uncertainties in the determination of nhave not been considered. They will give rise to larger error limits for the wave numbers. Within the error limit then0:4-value of the azimuthal wave number, m, agrees with that one determined
using ground-based magnetic ®eld observations. This agreement strongly favours a drift-bounce resonance instability withn 0:4 as a source mechanism for the observed giant pulsation. Any waves generated in n ÿ0:4 resonance would barely be observable at the ground due to the large m-value causing a large spatial attenuation of the wave signal below the ionosphere (e.g. Hughes and Southwood, 1976).
However, resonance condition (20) is only a neces-sary condition for instability. A more detailed investi-gation requires consideration of the physics of low-frequency ring current instabilities. General approaches to this problem have been provided by Southwood (1976) and Chen and Hasegawa (1991). While South-wood (1976) has studied the problem using a drift-kinetic approach, Chen and Hasegawa (1991) make use of the more detailed gyro-kinetic approximation. In the following we shall argue that the observed giant pulsation is generated by the unstable high-energy bump-on-the-tail part of the proton distribution. For these resonant protons and an azimuthal wave number of the orderjmj= 30 the ratiok?q0:1 (where k? is
the transverse wave number and q the resonant proton gyroradius). Thus a drift-kinetic approximation as used by Southwood (1976) seems appropriate.
Southwood's (1976) general approach to low-fre-quency instabilities in the ring current provides an elegant framework for studying wave generation in the ring current. His central Eq. (22) may be written as (see also Chen and Hasegawa, 1991)
x2dK 2 dU dM ix dR 23
where xis the wave frequency,x2dK the change in ¯ux
tube integrated plasma kinetic energy,dU the change in plasma bulk energy, and dM the perturbation of magnetic energy. The term ixdR describes energy changes due to resonant wave-particle interactions with
dRgiven in a dipole ®eld topology by
dR X
n Z
ds
Z
2p2 j~BjdldW
vk janj 2
d xrÿmXDÿnXB dWdf : 24
Here the integration is over the ¯ux tube volume de®ning the interaction region, ~B is the background magnetic ®eld, f the particle distribution function, W the particle energy and l its magnetic moment. Sum-mation is over the dierent wave modes. The coecient
an describes the Fourier expansion of the particle
bounce motion along ®eld lines (Southwood, 1976). The delta-distribution describes the resonance character of the interaction.
Equations such as Eq. (23) are very suitable to analyse a plasma's stability. For example, without resonant particlesdR0, we have
x2dK 2 dU dM 25
and the plasma is unstable, if dUdM<0, as in this casexbecomes an imaginary number. The mirror and ®rehose modes are instabilities of this kind, i.e. they are purely growing modes (Kruskal and Oberman, 1958; Chen and Hasegawa, 1991). In more general cases with
xxricfrom the imaginary part for the growth rate
we assume:
c 2ddRK : 26
Thus an instability may be driven by resonant wave-particle interaction, provideddR>0 anddK>0.
The original treatment by Southwood (1976) discuss-es the eect of an adiabatically injected background plasma population and concludes that mainly particles in n ÿ1 resonance destabilise the plasma, while the n 1 particles have a stabilising in¯uence. Thus, Southwood (1976) as well as Green (1979), Hughes et al. (1978), Glassmeier (1980), Poulter et al. (1983), or Chisham and Orr (1991) assumed that only even-mode harmonics with an asymmetric electric ®eld variation can be excited. As some observations suggest that giant pulsations are odd-mode ®eld line eigenoscillations (with symmetric electric ®eld variation) the bounce-resonance instability has been ruled out as a viable generation mechanism for these peculiar wave modes (Green 1976; Takahashiet al., 1992). However, in view of the clear correlation found between the October 4, 1978 giant pulsation event and substorm associated proton injection as well as the observation of a bump-on-the-tail distribution of the ring-current plasma a somewhat dierent interpretation emerges, if one care-fully considers the assumptions on which the even-mode hypothesis is based.
In particular, Southwood (1976) assumed an adia-batically injected background plasma distribution as he is mainly interested in studying the stability of the background plasma against bounce resonance interac-tions. In the particular case we are analysing the background plasma distribution has been changed by substorm injected protons. The total distribution func-tion may thus be approximated as
f l;W fB l;W fI l;W 27
where indices `B' and `I' denote the background distribution and the substorm injected particles, respec-tively. The gradient df=dW may be estimated as
df
dW
@fB
@W jl;L @W@fIjl;L @W@L @f@LBjl;W 28
Here, the variation of the substorm injected particle distribution fI l;W with radial distance has been
Due to a lack of detailed information about the substorm injected distribution we assume that it varies with energyW in a manner proportional to the variation of the background plasma. Thisansatz
@fI
@W jl;L g@W@fBjl;L 29
for the phase space gradient of the bump-on-the-tail part of the proton distribution is very convenient as it allows formulation of the gradient in terms of the drift frequencyXD and via the resonance condition in terms of the bounce frequency XB, the mode number n, and the azimuthal wave number m. The proportionality between between the background and bump-on-the-tail parts is described by the factorg.
The phase space gradient for the resonant particles of the background plasma can be approximated as (South-wood et al., 1969)
@fB
@W nXB
xr
@fB
@W l;L; 30
and the total phase space gradient is approximately given by the expression,
df
dW
g nXB
xr @fB
@W l;L: 31
The growth rate Eq. (26) then reads
cdK X
n Z
ds
Z
p2 j~BjdldW
vk janj 2
d xrÿmXDÿnXB
g nxXB r
@fB
@W l;L: 32
Under normal adiabatic conditions @fB=@W < 0, and
the ring current plasma becomes unstable against resonant wave-particle interaction if
g nXB
xr < 0 33
If there are no substorm injected particles present, i.e.
g0, nXB
xr < 0 34
results, which gives onen<0 for instability. Particles in n>0 resonance are obviously stabilising the plasma. However, if g60, as expected for a bump-on-the-tail distribution, we have
g < ÿnXBx
r 35
for instability. Particles in n>0 resonance now con-tribute to instability provided jgj>jnXB
xrj, that is if the
bump-on-the-tail distribution phase space gradient is suciently large. The interpretation of this ®nding is straightforward. Instability is due to two dierent gradients, the one in L-space and the one in W-space. With respect to theL-space gradient background plasma particles in n>0 resonance stabilise the plasma, while bump-on-the-tail n>0 resonant particles destabilise
due to their positive W-space gradient. If the later particles outweigh the stabilising eect of the back-ground particles, that is if the bump-on-the-tail is strong enough, the plasma becomes unstable.
In the present case xr 6:2810ÿ2 rad sÿ1, XB10ÿ1 rad sÿ1, and n0:45. Thus, a value
g<ÿ0:72 would be sucient for instability of the odd-harmonic such as the n0:45 mode. Such a gradient is compatible with the GEOS-2 particle distri-bution observations. This discussion of the sucient instability conditions therefore provides further support of a bump-on-the-tail driven bounce-resonance instabil-ity driving the observed Pg.
A question remaining is the generation of the the unstable phase space density, that is the bump-on-the-tail distribution causing @f=@W >0. Hughes et al. (1978) argue that unstable bump-on-the-tail distribu-tions are naturally generated due to the combined action of gradient and curvature drift as well as~E~Bdrift due to a solar wind induced dawn-dusk electric ®eld. The ~E~B drift is essentially energy independant and larger than the gradient drift for energies below about 10 keV. Above 10 keV ions gradient drift from the afternoon side westward via the frontside to the morning side (see Fig. 8). The ~E~B drift opposes the gradient drift of protons, giving rise to a velocity ®lter eect allowing only protons above 10 keV to drift westward to the morning.
In the present case associated proton energies are much larger, and ~E~B drift eects are essentially unimportant. Velocity dispersion will lead to unstable distribution far away from the injection point as schematically shown in Fig. 8. Faster protons will reach a certain point earlier than lower energy particles. High-energy particles are then added to the local background plasma at a higher rate than low-energy particles. This causes the build-up of the bump-on-the-tail distribution. This distribution, however, is not stable as lower energy particles follow up and the higher energy ones are moving further. Only if the dispersion is large enough and the injection of particles lasts for a long enough time
can an unstable distribution develop giving rise to the observed wave activity.
A detailed examination of how the unstable distribu-tion evolves is beyond the scope of the present work and will be the subject of a future study on proton drifts in a realistic magnetic ®eld geometry. However, several peculiarities of giant pulsations may be explained by this source hypothesis. For example, the instability can only set in at a great distance from the injection point, where the lifetime of the unstable distribution is larger than the inverse of the growth rate of the instability. The distri-bution is not only relaxing back to equilibrium by wave-particle interactions, but also due to the lower energy particles arriving and ®lling the low energy part of the phase space. This condition for the instability to occur ®ts very well with the observations that giant pulsations are usually observed in the early morning hours, that is at a great distance away from the proton injection point.
5 Conclusions
A ground-satellite coordinated case study of a giant pulsation event has been presented, which demonstrates that giant pulsations are associated with a drifting proton cloud injected into the nightside magnetospheric ring current due to substorm activity. The following conclusions are reached based on results from this study and previous work:
1. Substorm activity in the nightside magnetosphere causes injection of high-energy protons from the mag-netotail into the eveningside magnetosphere.
2. Under the in¯uence of gradient and curvature drifts these particles drift westward towards the dayside in the ring current regime. Velocity dispersion occurs, much as observed.
3. This velocity dispersion can lead to bump-on-the-tail phase space distributions with the injected protons overlaying the ambient plasma. Further theoretical work and numerical studies are required to follow the evolution of the injected particle population in detail. It may be speculated that dispersion must be large enough to result in distributions unstable against wave-particle interaction. The predominantly morningside occurrence of Pg's could be explainable in this way.
4. The bump-on-the-tail distribution causes a positive phase space density gradient, that is @f=@W > 0. Under assumptions discussed in more detail earlier this positive gradient causes the plasma to become unstable against the excitation of odd-mode type fundamental ®eld line eigenoscillations via a drift-bounce resonance. The conclusion of an odd-mode being excited diers from previous studies (e.g. Southwood, 1976; Chen and Hasegawa, 1991) where the unstable plasma has been assumed to be an adiabatically generated type with @f=@W < 0. The addition of particles injected due to substorm activity changes the phase space gradient and allows the odd-mode to become unstable. For the ®rst time this odd-mode structure has been con®rmed observationally for the October 4, 1978 event, discussed in this study.
5. A resonant wave-particle interaction also allows us to understand the typical quasi-monochromatic wave form of Pgs. Furthermore, resonance requires a signi®cant Doppler shift, which causes a large azimuthal wave number, m, of any unstable wave. Observed and theoretically deduced m-values for the event under discussion agree fairly well.
6. A large azimuthal wave number, aboutm ÿ26 in our case, requires the excited wave to propagate as a guided poloidal mode. These modes have a predominantly radial magnetic ®eld perturbation as ®rst discussed by Dungey (1954). A radial magnetic ®eld variation is associated with an azimuthal electric ®eld oscillation. The observed electric ®eld perturbation at GEOS-2 is mainly in the azimuthal direction, in accord with the hypothesis. Furthermore, at the ground a magnetic ®eld oscillation is expected which is rotated by about 90against that in
the magnetosphere (e.g. Hughes, 1974; Glassmeier, 1984). Thus the Pg oscillation should be largest in the D component at the ground. This is as observed.
Finally, we would like to mention that giant pulsa-tions are evidence of a very fundamental process for the generation of unstable particle phase space distributions in the Earth's magnetosphere.
Acknowledgements. We are grateful to Gordon Rostoker for providing us with Fort Providence magnetic ®eld observatories. The work by K.H.G. and S.B. was supported by the Deutsche Forschungsgemeinschaft and the German Space Agency DARA (now DLR), the work by A.K. by the German Bundesministerium fuÈr Wissenschaft und Technologie, and the work of U.M. through a Werner-Heisenberg-Stipendium of the Deutsche Forschungs-gemeinschaft. Support from Frank Budnik, Stefan Markgraf and Nicole Cornilleau-Wehrlin is gratefully acknowledged.
The Editor in chief thanks T.K Yeoman and another referee for their help in evaluating this paper.
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